Several algorithms solve exactly or approximately the OT problem between two discrete measures.
Network flow simplexes [1]} solve OT for any inputs \((,)\)  in \(O(N^3\log N)\)  time.
When measures have the same cardinality \(N\)  and uniform weights \(1/N\) , the Hungarian algorithm [2]} and the auction algorithm [3]}, [4]} compute the permutation matrix in \(O(N^3)\)  time.
These methods are efficient for medium scale problems, where typically \(N\lesssim 10^4\) .
